Question

nl and mk are diameters of ⊙t.
identify whether ( moverarc{jl} ) is a major arc, minor arc, or semicircle.
find its measure.

Explanation:

Step1: Determine the type of arc

A minor arc is less than \(180^\circ\), a major arc is more than \(180^\circ\), and a semicircle is \(180^\circ\). First, find the measure of \(\overset{\frown}{JL}\) by adding the central angles. The central angles for \(\angle JTN = 88^\circ\), \(\angle NTL = 180^\circ - 88^\circ - 44^\circ - 48^\circ\)? Wait, no, let's look at the given angles. The angles at center \(T\): \(\angle NTJ = 88^\circ\), \(\angle JT K = 44^\circ\), \(\angle KTL = 48^\circ\)? Wait, no, actually, \(NL\) and \(MK\) are diameters, so \(\angle NTL = 180^\circ\) (since diameter makes a straight angle). Wait, the angles given: \(\angle NTJ = 88^\circ\), \(\angle JT K = 44^\circ\), \(\angle KTL = 48^\circ\)? Wait, no, let's sum the angles for arc \(JL\). The arc \(JL\) is formed by the central angles \(\angle JT K\), \(\angle KTL\), and wait, no. Wait, \(NL\) is a diameter, so \(\angle NTL = 180^\circ\). The angle \(\angle NTJ = 88^\circ\), \(\angle JT K = 44^\circ\), so then \(\angle KTL = 180^\circ - 88^\circ - 44^\circ = 48^\circ\)? Wait, maybe better to calculate the measure of arc \(JL\) by adding the central angles. The central angles for arc \(JL\) are \(\angle JT K = 44^\circ\) and \(\angle KTL = 48^\circ\)? No, wait, arc \(JL\) is from \(J\) to \(L\), so the central angles are \(\angle JT K\) (44°), \(\angle KTL\) (48°), and wait, no, maybe I misread. Wait, the angles at \(T\): \(\angle NTJ = 88^\circ\), \(\angle JT K = 44^\circ\), and since \(MK\) is a diameter, \(\angle MTK = 180^\circ\). Wait, maybe the measure of arc \(JL\) is the sum of \(\angle JT K\) (44°) and \(\angle KTL\) (48°) and wait, no, let's check the straight line. \(NL\) is a diameter, so the angle from \(N\) to \(L\) through \(T\) is \(180^\circ\). The angle from \(N\) to \(J\) is \(88^\circ\), \(J\) to \(K\) is \(44^\circ\), so \(K\) to \(L\) is \(180^\circ - 88^\circ - 44^\circ = 48^\circ\). Then arc \(JL\) is from \(J\) to \(L\), so the central angle is \(\angle JTL\), which is \(\angle JT K + \angle KTL = 44^\circ + 48^\circ = 92^\circ\)? Wait, no, that can't be. Wait, maybe I made a mistake. Wait, \(NL\) is a diameter, so \(\angle NTL = 180^\circ\). The angle \(\angle NTJ = 88^\circ\), so the angle from \(J\) to \(L\) through \(T\) is \(\angle JTL = 180^\circ - 88^\circ = 92^\circ\)? No, that's not right. Wait, no, the arc \(JL\): let's see the points. \(J\), \(K\), \(L\), \(M\), \(N\). \(NL\) is a diameter, so \(N\) and \(L\) are opposite. \(MK\) is a diameter, so \(M\) and \(K\) are opposite. So the central angles: \(\angle NTJ = 88^\circ\), \(\angle JT K = 44^\circ\), so \(\angle KTL = 180^\circ - 88^\circ - 44^\circ = 48^\circ\) (since \(N\) to \(L\) is 180°). Then arc \(JL\) is from \(J\) to \(L\), so the central angle is \(\angle JTL = \angle JT K + \angle KTL = 44^\circ + 48^\circ = 92^\circ\)? Wait, but 92° is less than 180°, so it's a minor arc. Wait, no, wait, maybe the arc \(JL\) is actually the sum of \(\angle NTJ\) (88°) and \(\angle NTL\) is 180°, no. Wait, let's start over. The measure of an arc is equal to the measure of its central angle. So for arc \(JL\), the central angle is \(\angle JTL\). To find \(\angle JTL\), we can use the fact that \(NL\) is a diameter, so \(\angle NTL = 180^\circ\). The angle \(\angle NTJ = 88^\circ\), so \(\angle JTL = 180^\circ - 88^\circ = 92^\circ\)? No, that's not considering the other angle. Wait, no, the angles at \(T\): \(\angle NTJ = 88^\circ\), \(\angle JT K = 44^\circ\), so \(\angle KTL = 180^\circ - 88^\circ - 44^\circ = 48^\circ\). Then \(\angle JTL = \angle JT K + \angle KTL = 44^\circ + 48^\circ = 92^\circ\). Since 92° is less than 180°, it's a minor arc. Wait, but maybe I messed up the angles. Wait, the problem is to find the measure of arc \(JL\). Let's sum the central angles: \(\angle NTJ = 88^\circ\), \(\angle JT K = 44^\circ\), and then from \(K\) to \(L\) is 48°? Wait, no, maybe the arc \(JL\) is composed of \(\angle JT K\) (44°), \(\angle KTL\) (48°), and wait, no, \(NL\) is a diameter, so the total around \(T\) is 360°, but diameters divide into 180° each. Wait, \(NL\) is a diameter, so the arc \(NL\) is 180°. The arc \(JL\) is from \(J\) to \(L\), so the central angle is \(\angle JTL\). Let's calculate \(\angle JTL\): \(\angle NTJ = 88^\circ\), so \(\angle JTL = 180^\circ - 88^\circ = 92^\circ\)? No, that's not right. Wait, maybe the angles given are \(\angle NTJ = 88^\circ\), \(\angle JT K = 44^\circ\), and \(\angle KTL = 48^\circ\), so adding those: 88 + 44 + 48 = 180? Wait, 88 + 44 is 132, 132 + 48 is 180. Oh! Wait, \(NL\) is a diameter, so the sum of angles from \(N\) to \(J\) to \(K\) to \(L\) is 180°. So \(\angle NTJ = 88^\circ\), \(\angle JT K = 44^\circ\), \(\angle KTL = 48^\circ\), and 88 + 44 + 48 = 180. So arc \(JL\) is from \(J\) to \(L\), which is the sum of \(\angle JT K\) (44°) and \(\angle KTL\) (48°)? No, wait, \(J\) to \(L\) is \(J\) to \(K\) to \(L\), so the central angle is 44° + 48° = 92°? Wait, no, 88 + 44 + 48 = 180, so \(N\) to \(L\) is 180°, so \(J\) to \(L\) is \(N\) to \(L\) minus \(N\) to \(J\), which is 180° - 88° = 92°. Yes, that's correct. So the measure of arc \(JL\) is 92°, which is less than 180°, so it's a minor arc.

Step2: Calculate the measure

The measure of arc \(JL\) is the sum of the central angles or the difference from 180° (since \(NL\) is a diameter). Since \(\angle NTJ = 88^\circ\), and \(NL\) is a diameter (so arc \(NL\) is 180°), then arc \(JL\) = 180° - 88° = 92°? Wait, no, that's not considering the other angle. Wait, no, the central angle for arc \(JL\) is \(\angle JTL\), which is equal to \(\angle JT K + \angle KTL\) = 44° + 48° = 92°, which matches. So the measure is 92°, and since 92° < 180°, it's a minor arc.

Answer:

First, identify the type: minor arc (since 92° < 180°). Then the measure: 92°
So the type is minor arc, and the measure is \(\boxed{92}\) (for the measure) and the type is minor arc. But the question has two parts: identify (minor arc) and find measure (92°). So for the first box: minor arc, second box: 92.